Navier stokes equeations, shear term

[navalstudent]

Hey!, I was repeating for myself a course I had from a earlier year, fluid mechanics. I looked at the derivation of the navier stokes equations, and there is one term that does not give meaning to me.


Take a look at the x-momentum equation here:
http://www.grc.nasa.gov/WWW/K-12/airplane/nseqs.html

The term I don't get is the d/dx(tau_xx). I mean what does this term mean physically? It is a shear stress in the x-direction that is acting on the the surface wit an x-normal? Tau_xy and tau_xz is easy to understand from ordinary mechanics, but not tau_xx. I tought only the term -d/dx(P) would give a normal stress in the x-direction.

So can someone please explain to me how we can have a shear stress in the x-direction acting on the x surface(y-z-plane).

 

[Andy Resnick]

That is an odd way to write the equations; I thought the diagonal components of the stress tensor (tau_xx, tau_yy, tau_zz) end up in the pressure term.

 

[navalstudent]

That is an odd way to write the equations; I thought the diagonal components of the stress tensor (tau_xx, tau_yy, tau_zz) end up in the pressure term.

Hello Andy Resnick.

That is what I also tought at first, but the fact is that these terms are not the classical preassure terms. And I still have not been able to find a physical explanation for them.

Could one say that since the fluid has viscosity, we will have a "glue-effect", so that the normal strains contains preassure terms, and since the fluid is sticky, the viscosity will "drag" the fluid forward?(if we have a velocity gradient in the normal-direction). Note: I have not talked about shear-stresses here.

 

[Pythagorean]

here's what I dug up:

keywords: Turbulence modeling, dynamic viscocity (aka absolute viscocity), reynold's stress

It should also be noted that the theory of the Reynolds stress is quite analogous to the kinetic theory of gases, and indeed the stress tensor in a fluid at a point may be seen to be the ensemble average of the stress due to the thermal velocities of molecules at a given point in a fluid. Thus, by analogy, the Reynolds stress is sometimes thought of as consisting of an isotropic pressure part, termed the turbulent pressure, and an off-diagonal part which may be thought of as an effective turbulent viscosity.

http://en.wikipedia.org/wiki/Reynolds_stress

Dynamic Viscocity:
http://www.engineeringtoolbox.com/dynamic-absolute-kinematic-viscosity-d_412.html

 

[Andy Resnick]

I tried parsing my go-to reference for this stuff (Non-Linear field theories of Mechanics, Handbuch der Physics vol III/3) and was promptly confused.

They do write down a general constitutive relation for fluids T + p1, and there is no restriction on the Cauchy stress tensor, but by the time they get to Korteweg's theory, I got lost in a maze of tensor representation theorems.

I wish the GRC site gave a little more information, instead of just tossing out a formula.